As this kind of apparatus, there is conventionally known in Japanese Published Examined Patent Application No. 709/1975 an apparatus in which, at the time of gear engagement or gear meshing due to changing over from a neutral range to a forward range, the speed-change stage of the automatic transmission is first set to a predetermined high-speed stage and is then down-shifted to a low-speed stage, to prevent a large driving force by the low-speed stage from being suddenly transmitted, thereby reducing the shock at the time of gear engagement. In this prior art, in order to securely perform the down-shifting via the high-speed stage at the time of gear engagement, an arrangement is made so that the high-speed stage is maintained until a predetermined time has elapsed from the time of gear engagement, out of consideration of the time required for engagement of frictional engaging elements which establish the above-described predetermined high-speed stage.
However, the time required for the engagement of the frictional engaging elements varies with mechanical fluctuations of the automatic transmission, oil temperature, rotational speed of an engine, hydraulic engaging characteristics of the frictional engaging elements, or the like. Therefore, if the time for holding the speed-change stage to the high-speed stage is kept constant as described above, there may occur a case in which the automatic transmission is down-shifted before the frictional engaging elements for the high-speed stage are engaged, with the result that the shock at the time of gear engagement are not sufficiently reduced. On the other hand, if the time for holding the speed-change stage to the high-speed stage is prolonged, the down-shifting will be delayed, resulting in a poor starting characteristics.
Further, in Japanese Published Examined Japanese Patent Application No. 69018/1991, there is known an apparatus in which, at the time of gear engagement, an automatic transmission is maintained at a high-speed stage until the rotational speed of a turbine of a torque converter is down to a predetermined value. However, since the rotational speed of the turbine varies with the rotational speed of the engine, or the like, the frictional engaging elements for the high-speed stage may not necessarily be sufficiently engaged when the rotational speed of the turbine is down to the predetermined value. Further, since the engaging conditions of the frictional engaging elements at the time when the rotational speed of the turbine has lowered to the predetermined value are different between the time when the vehicle is completely stopped and the time when it is not, the time of engagement of the frictional engaging elements for the high-speed stage cannot accurately be determined from the rotational speed of the turbine alone.